Equations For Quadratic Regression Does anyone know the specific equations for the three parameters in a least-squares quadratic regression? I'm looking for something like $\beta_1=,\beta_2=,\beta_3=$ for each of $y=\beta_1+\beta_2x+\beta_3x^2$. To be clear, the right side of each of these equations should be evaluateable, using the data, to find the parameter. I was able to find the equations for linear regression on line, but google hasn't turned anything up for this.
Thanks in advance
 A: Assuming $y_j =\beta_1 +\beta_2 x_j +\beta_3 x_j^2 + \epsilon_j$ for $j = 1\dots n$, we can write the matrix equation
$$\mathbf{Y}= \mathbf{XB} + \mathbf{E}$$
where
$$\mathbf Y = [y_1 \ y_2 \ \dots \ y_n]^T,$$
$$
   \mathbf X = 
   \left[ \begin{array}{c}
      1      & x_1    & x_1^2 \\
      1      & x_2    & x_2^2 \\
      \vdots & \vdots & \vdots \\
      1      & x_n    & x_n^2 \\
   \end{array} \right],
$$
$$\mathbf B = [\beta_1 \ \beta_2 \ \beta_3]^T,$$
and
$$\mathbf E = [e_1 \ e_2 \ \dots \ e_n]^T$$
We seek the vector $\mathbf B$ that minimizes
$$e_1^2 + e_2^2 + \cdots + e_n^2 = \mathbf E^T \mathbf E$$
Assuming that $\mathbf X^T \mathbf X$ is nonsingular (usually a safe assumption), we find that
$$\mathbf B = (\mathbf X^T \mathbf X)^{-1} \mathbf X^T \mathbf Y$$
A: For $y =\beta_1 +\beta_2 x +\beta_3 x^2$, let's define
$$x_1 = x$$
and
$$x_2= x^2.$$
Now we can use the equations of multiple linear regression:
$$S_{11}= \sum_{n=1}^N x_1^2- \frac{(\sum_{n=1}^N x_1)^2}{N}$$
$$S_{12}= \sum_{n=1}^N x_1x_2- \frac{(\sum_{n=1}^N x_1\sum_{n=1}^N x_2)}{N}$$
$$S_{22}= \sum_{n=1}^N x_2^2- \frac{(\sum_{n=1}^N x_2)^2}{N}$$
$$S_{y1}= \sum_{n=1}^N y x_1- \frac{(\sum_{n=1}^N y\sum_{n=1}^N x_1)}{N}$$
$$S_{y2}= \sum_{n=1}^N y x_2- \frac{(\sum_{n=1}^N y\sum_{n=1}^N x_2)}{N}$$
$$\overline{x}_1 = \frac{(\sum_{n=1}^N x_1)}{N}$$
$$\overline{x}_2 = \frac{(\sum_{n=1}^N x_2)}{N}$$
$$\overline{y} = \frac{(\sum_{n=1}^N y)}{N}$$
$$\beta_2=\frac{S_{y1}S_{22}-S_{y2}S_{12}}{S_{22}S_{11}-S_{12}^2}$$
$$\beta_3=\frac{S_{y2}S_{11}-S_{y1}S_{12}}{S_{22}S_{11}-S_{12}^2}$$
$$\beta_1=\overline{y}-\beta_2\overline{x}_1-\beta_3\overline{x}_2$$
